In Hamlet Act 1 Scene 2, what are the relationships that characters have to one another?

In order to understand the rest of the play, you must
understand the relationships the characters have with one another, and you must
understand the dynamics within those relationships.  In Scene 2 we first meet King
Claudius who has taken over the throne of his recently deceased brother.  He has also
just recently married his brother's wife/widow, Gertrude.  Prince Hamlet is the son of
Gertrude and the former king, so that makes Claudius his uncle and how his step father. 
Hamlet is not happy about this turn of events and his first line reflects this
attitude.  Claudius call Hamlet his son, and Hamlet, in an aside, says, "a little more
than kin and less than kind."  He feels a little too closely
related to Claudius now. Hamlet loves his mother but is diappointed in her
decision to marry Claudius.  He agrees to stay at Elisinore for her sake.  This shows
that Hamlet is still a dutiful son at heart. Claudius, in an attempt to show fatherly
concern, gives a very long speech to Hamlet about how he needs to more quickly accept
his father's death as an event that is natural and expected and therefore should be
grieved and moved past.  He comes across as rather cold and practical, rather than
loving and understanding.

In this scene we also briefly
meet characters who become more important later in the play.  Laertes is the son of
Polonius, and Polonius is a chief courtier and adviser to Claudius.  Laertes goes
through the public obligation of asking the King's permission to return to college now
the the funeral and wedding are over.  Claudius, wanting to maintain the support of a
courtier like Polonius, is very flattering in his behavior towards  Laertes , and grants
the request.

This scene also shows us the friendship
between Hamlet and Horatio.  In scene 1, Horatio is portrayed as the learned man
who verifies the appearance of the ghost.  In this scene, we see the friendship between
the two men and how supportive Horatio is light of all of the change in Hamlet's life. 
Horatio, as kindly as possible, and with a lot of patience of Hamlet's questions,
reveals all of the details he can about the appearance of the ghost of King Hamlet.  He
is rightly concerned about Hamlet's wanting to see it for himself, but he is supportive
nonetheless.